2003 Indonesia MO Problems
Contents
Day 1
Problem 1
Prove that is divisible by for every integers .
Problem 2
Given a quadrilateral . Let , , , and are the midpoints of , , , and , respectively. and intersects at . Prove that and .
Problem 3
Find all real solutions of the equation .
[Note: For any real number , is the largest integer less than or equal to , and denote the smallest integer more than or equal to .]
Problem 4
Given a matrix, where each element is valued or . Let be the product of all elements at the row, and be the product of all elements at the column. Prove that:
Day 2
Problem 5
For every real number , prove the following inequality
and determine when the equality holds.
Problem 6
A hall of a palace is in a shape of regular hexagon, where the sidelength is . The floor of the hall is covered with an equilateral triangle-shaped tile with sidelength . Every tile is divided into congruent triangles (refer to the figure). Every triangle-region is colored with a certain color so that each tile has different colors. The King wants to ensure that no two tiles have the same color pattern. At least, how many colors are needed?
Problem 7
Let be positive integers such that and the greatest common divisor of and is . Prove that if divides , then .
Problem 8
Given a triangle with as the right angle, and the sidelengths of the triangle are integers. Determine the sidelengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.
See Also
2003 Indonesia MO (Problems) | ||
Preceded by 2002 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2004 Indonesia MO |
All Indonesia MO Problems and Solutions |